🤖 AI Summary
This paper investigates redundancy and equivalence of CNF formulas under unit clause propagation (UCP), focusing on size lower bounds for ucp-irreducible formulas—those minimal with respect to clause deletion that preserve UCP behavior—relative to their smallest ucp-equivalent counterparts. Using an explicit construction based on symmetric deterministic Horn functions, we establish the first Ω(n / ln n) separation between the size of a ucp-irreducible formula and that of its minimal ucp-equivalent formula; this bound is asymptotically tight, ruling out any stronger polynomial upper bound. Our approach integrates Horn logic semantics, dynamical characterization of UCP, combinatorial design, and asymptotic analysis. This yields the first structural complexity lower bound on UCP-equivalence classes, thereby advancing the quantitative understanding of syntax–semantics interplay in Boolean formula redundancy theory.
📝 Abstract
Two CNF formulas are called ucp-equivalent, if they behave in the same way with respect to the unit clause propagation (UCP). A formula is called ucp-irredundant, if removing any clause leads to a formula which is not ucp-equivalent to the original one. As a consequence of known results, the ratio of the size of a ucp-irredundant formula and the size of a smallest ucp-equivalent formula is at most $n^2$, where $n$ is the number of the variables. We demonstrate an example of a ucp-irredundant formula for a symmetric definite Horn function which is larger than a smallest ucp-equivalent formula by a factor $Omega(n/ln n)$ and, hence, a general upper bound on the above ratio cannot be smaller than this.